Embed Size (px)
Citation preview
University of Nebraska - LincolnDigitalCommons@University of Nebraska - LincolnFaculty Publications from the Department ofElectrical and Computer Engineering Electrical & Computer Engineering, Department of
2015
Robust Optimization for Bidirectional DispatchCoordination of Large-Scale V2GXiaoqing BaiUniversity of Nebraska-Lincoln, [email protected]
Wei QiaoUniversity of Nebraska-Lincoln, [email protected]
Follow this and additional works at: http://digitalcommons.unl.edu/electricalengineeringfacpub
Part of the Computer Engineering Commons, and the Electrical and Computer EngineeringCommons
This Article is brought to you for free and open access by the Electrical & Computer Engineering, Department of at DigitalCommons@University ofNebraska - Lincoln. It has been accepted for inclusion in Faculty Publications from the Department of Electrical and Computer Engineering by anauthorized administrator of DigitalCommons@University of Nebraska - Lincoln.
Bai, Xiaoqing and Qiao, Wei, "Robust Optimization for Bidirectional Dispatch Coordination of Large-Scale V2G" (2015). FacultyPublications from the Department of Electrical and Computer Engineering. 261.http://digitalcommons.unl.edu/electricalengineeringfacpub/261
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
IEEE TRANSACTIONS ON SMART GRID 1
Robust Optimization for Bidirectional DispatchCoordination of Large-Scale V2G
Xiaoqing Bai and Wei Qiao, Senior Member, IEEE
Abstract—This paper proposes a robust optimization (RO)model for bidirectional dispatch coordination of large-scale plug-in electric vehicles (PEVs) in a power grid in which the PEVsare aggregated to manage. The PEV aggregators are consid-ered as a type of dispatchable demand response and energystorage resource with stochastic behaviors, and can supply loador provide ancillary services such as regulation reserve to thegrid. The proposed RO model is then reformulated as a mixed-integer quadratic programming model, which can be solvedefficiently. Computer simulations are performed for a powergrid with ten generators and three PEV aggregators to vali-date the economic benefit of the RO model for bidirectionaldispatch coordination of the PEVs and the robustness of theRO model to the uncertainty of the PEVs’ stochastic mobilitybehaviors.
Index Terms—Bidirectional dispatch, coordination, plug-inelectric vehicle (PEV), robust optimization (RO), smart grid,vehicle to grid (V2G).
NOMENCLATURE
Indices
i Index of thermal generators.j Index of plug-in electric vehicle (PEV) aggregators.t Index of time intervals during a dispatch period.T Total number of intervals for a dispatch period.n Number of thermal generators.m Number of PEV aggregators.
Parameters
a1i, a2i, a3i Fuel cost coefficients of thermal generator i.σ t Number of hours during each dispatch interval t.pec,t Capability unit price of reserve during t.LMPj,t Locational marginal price (LMP) of PEV aggre-
gator j during t.kup
i /kdowni Start-up/down cost of thermal generator i.
Ploadt Base load without PEV aggregators during t.
Pi/Pi Lower/upper active power limit of generator i.Rdown
i /Rupi Ramp-down/up limit of generator i.
Manuscript received May 11, 2014; revised October 20, 2014 andJanuary 8, 2015; accepted January 20, 2015. This work was supportedin part by the U.S. National Science Foundation under CAREER AwardECCS-0954938, and in part by the National Natural Science Foundation ofChina under Grant 51367004. Paper no. TSG-00401-2014.
The authors are with the Power and Energy Systems Laboratory,Department of Electrical and Computer Engineering, University ofNebraska–Lincoln, Lincoln, NE 68588-0511 USA (e-mail: [email protected];[email protected]).
Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSG.2015.2396065
Hdowni /Hup
i Minimum down/up time limit of generator i.PORi,t Preference of generator i to provide reserve
during t (1 if it prefers; otherwise 0).P
chgj /P
disj Limits of charging/discharging power of PEV
aggregator j, which are determined by the max-imum charging/discharging power of the PEVsin the aggregator as well as the physical inter-face (e.g., chargers and power line capacities)between PEV aggregator j and the power grid.
PIOj,t Status of PEV aggregator j to participate in gridoperation to charge, discharge to supply load, orprovide regulation reserve during t.
ηchgj /ηdis
j Charging/discharging efficiency of PEV aggre-gator j.
wSoCj,t Maximum energy capacity of PEV aggregator j
during t.wSoC
j,t Predicted available energy capacity of PEVaggregator j during t.
EConj The total energy consumed by aggregator j for
driving in T .E
SoCj Maximum energy capacity of PEV aggregator j.
Rmin General minimum active power requirementof a regulation provider specified by the gridoperator.
SR Reserve requirement.τ Maximum time allowed for a thermal genera-
tor to ramp up to deliver the committed reservecapacity.
Cost Variables
CFt Total fuel cost of all thermal generators during t.CRt Total cost of thermal generators to provide reserve
service during t.COt Total cost of starting up and shutting down thermal
generators during t.PRt Total cost for PEV aggregators to provide the regula-
tion reserve service during t.PSt Total cost for PEV aggregators to discharge power to
the power grid during t.
Decision Variables
Pi,t Active power of thermal generator i during t.ui,t Unit commitment (UC) status (0/1) of thermal
generator i during t.oi,t/vi,t Unit start-up/shut-down status (0/1) of thermal
generator i during t.
1949-3053 c© 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
2 IEEE TRANSACTIONS ON SMART GRID
Pchgj,t /Pdis
j,t Charging/discharging power of PEV aggregatorj during t.
zchgj,t /zdis
j,t Charging/discharging status (0/1) of PEV aggre-gator j during t.
Rcj,t/Rd
j,t Reserve capacity of PEV aggregator j in thecharging/discharging mode during t.
zcResj,t /zdRes
j,t Status (0/1) of PEV aggregator j in thecharging/discharging mode to provide regulationreserve service during t.
Uncertainty-Related Variables
wSoCj,t Uncertain available energy capacity of PEV aggrega-
tor j during t.γj,t Degree of uncertainty of the available energy capacity
of PEV aggregator j during t and |γj,t| ≤ 1.RSoC
j Uncertainty set of wSoCj for aggregator j.
Uncertainty-Related Parameters
α Degree of variation of wSoCj,t from wSoC
j,t .�j Budget of uncertainty of the PEV aggregator j, which is
the maximum level of uncertainty of the available energycapacity that the PEV aggregator j is willing to tolerate,where 1/T‖γj,t‖1 ≤ �j, t = 1, . . . , T, 0 ≤ �j ≤ 1.
I. INTRODUCTION
ECONOMIC and environmental incentives and advancesin technology are driving dramatic changes in modern
electric power grid. One of the changes are PEVs, which willoffer customers a promising way to save gasoline cost, andhelp reduce carbon emissions as well as other pollutants. Inaddition to economic and environmental benefits, PEVs alsooffer a potential source of energy storage, which is valuable tothe electric power grid. The possibility of using PEV batteriesto provide ancillary services to support electric grids has beenstudied for more than one decade [1].
Although PEVs are still in the development stage, muchwork has been conducted to analyze the impact of PEVs onthe electric power grids from the energy, environmental, andeconomic points of view [2]–[4]. Through the vehicle to grid(V2G) technology, the parked PEVs within a certain area canconstitute a PEV aggregator when they are connected to thegrid through some intelligent equipment. Such a PEV aggre-gator can represent a well-defined responsive load and offeradditional generation capacity for the provision of ancillaryservices for the power grids [5], [6]. Some research has beenconducted on mathematical models and control algorithms foroperation of V2G and the associated power grids. For example,in [7], an optimal V2G aggregator was designed for frequencyregulation, where the dynamic programming method was usedto obtain the optimal charging control for each PEV. In [8],an optimal charging strategy for PEVs in a market environ-ment was obtained by using a quadratic programming method.In [9], the particle swarm optimization technique was used tominimize the fuel cost and emission in a power grid, wherethe V2G was operated either as load or energy storage.
However, [1]–[4] focused on creating mathematical modelsand control algorithms based on the traditional determin-istic optimization (DO)-based power dispatch frameworks
without considering the operating characters of the PEVaggregators [3], such as fast response, short duration, stochas-tic mobility of the PEVs within an aggregator, etc. Moreover,in most of the prior work, to make the models easy to solve,the active powers of the PEV aggregators were consideredas continuous variables and the energy losses were ignoredduring the charging and discharging operations. In practice,the charging and discharging operations of a PEV aggrega-tor cannot happen at the same time from the electric powergrid’s point of view. Furthermore, energy losses are alwaysinvolved in the charging and discharging processes of PEVsand the charging and discharging efficiencies may be different.To overcome these limitations, in this paper, the charging anddischarging powers of a PEV aggregator are represented bydifferent variables using mixed-integer constraints to betterhandle the aforementioned practical issues.
Based on the current technology, if a PEV has availablecharging/discharging capacity, it can be charged/dischargedwhen it is parked and connected to the grid throughsome charging/discharging equipment. Obviously, the avail-able charging/discharging capacity of a PEV aggregatordepends on the maximum energy capacity and the availableenergy capacity (i.e., the total usable energy stored in thePEV batteries) of the PEV aggregator as well as the stochasticmobility behaviors of the PEVs in the aggregator. Stochasticoptimization approaches have been used to model the V2Gdispatch problem in ancillary markets [6], [10], in which thestochastic behaviors of PEVs were modeled by using probabil-ity distributions. Unfortunately, it is usually difficult to identifyaccurate probability distributions for the uncertainties of theavailable charging/discharging capacities of a PEV aggregatordue to the lack of historical data. A scenario-based stochasticmodel was presented in [11] to solve the problem of operationcoordination of PEVs and wind power generation. To obtainan accurate solution, a large number of scenarios are required,which needs intensive computational cost.
In contrast to the stochastic optimization approaches, therobust optimization (RO) approach [12] does not model theprobability distribution, but only requires moderate informa-tion of the uncertainty, such as the mean and the upper andlower limits of the uncertainty. Furthermore, the optimal solu-tion generated from a RO model covers all realizations ofthe uncertainty over a specific set designed. Recently, the ROapproach has been applied to solve the problems of genera-tion expansion planning [13] and UC [14]–[16], in which theuncertainties of load demand and/or new energy resourceswere considered.
This paper extends the classical UC model to propose a DOmodel for bidirectional dispatch (i.e., charging and discharg-ing) coordination of large-scale V2G in a power grid. Basedon the DO model, an RO model is then proposed by construct-ing the available energy capacities of the PEV aggregators foreach dispatch interval as an uncertainty set according to thestochastic mobility behaviors of the PEVs. The proposed ROmodel is then reformulated as a deterministic mixed-integerquadratic programming problem, which can be solved effi-ciently by using an existing solver, such as CPLEX [17] orGurobi [18]. Simulation studies are conducted for a power
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
BAI AND QIAO: RO FOR BIDIRECTIONAL DISPATCH COORDINATION OF LARGE-SCALE V2G 3
grid with ten thermal generators and three PEV aggregators tovalidate the economic benefit and robustness of the proposedRO model.
II. PROPOSED RO MODEL FOR BIDIRECTIONAL DISPATCH
COORDINATION OF V2G
A. Problem Description
Consider a power grid with large-scale V2G. An entity whooperates PEV charging/discharging facilities and participatesin the grid operation, such as a distribution system operator,collects the operation preference of each PEV in a certainarea within its territory to generate an aggregated operationpreference for all the PEVs, i.e., a PEV aggregator, in that area.The size of the aggregator fleet depends on the population,the number of PEVs, and the number of charging/dischargingfacilities in the area. The information of the PEV aggregatoris shared with the grid operator, such as an independent sys-tem operator. The grid operator includes the PEV aggregatorsinto its resource commitment and economic dispatch routine,where each PEV aggregator is considered as a type of demandresponse and energy storage resource, but its primary func-tion is for transportation. This section proposes a RO-basedUC model for day-ahead (DA) operation planning of large-scale V2G in a power grid. First, a DO-based UC model isdesigned for dispatch coordination of the PEV aggregators.Next, the uncertainty sets of the problem are constructed.Based on the DO model and the uncertainty sets, the RO modelis formulated and is then converted to a deterministic mixed-integer quadratic programming problem, which can be solvedefficiently.
B. DO Model Formulation
When a PEV aggregator participates in the grid operation,it can absorb energy from or feed energy back to the grid forshifting the peak load or eliminating the violation of generatorramps. Furthermore, the available energy capacity of a PEVaggregator can be used to provide spinning/regulation reserveservice if necessary. Therefore, when participating in the gridoperation, a PEV aggregator has four operating modes: charg-ing, discharging to supply load, providing regulation reservein the charging status, and providing regulation reserve in thedischarging status. The four operating modes are representedby four binary (0/1) variables and the logical relations amongthem are formulated as the mixed-integer constraints in themodel. In this paper, the predicted regulation market capabilityprices and LMPs are used as the reserve capability unit pricesand discharging energy unit prices of the PEV aggregators,respectively.
The objective of the proposed model is to minimize thetotal cost of all thermal generators and all PEV aggregatorsin a given dispatch period T. The total cost of all thermalgenerators includes the costs of fuel (CFt), reserve capacityprovided (CRt), and operation (COt) during T. The total costof all PEV aggregators includes the costs for the reserve capac-ity provided (PRt) and discharging power (PSt) during T. Theproposed DO model is formulated by integrating the PEVaggregators into a classical UC model in [19] as follows,
where the objective function is
minT∑
t=1
CFt + CRt + COt + PRt + PSt (1)
where
CFt =n∑
i=1
a1iP2i,t + a2iPi,t + a3i
CRt = pec,t
n∑
i=1
PORi,t(min
(ui,t
(Pi − Pi,t
), ui,tτRup
i
))
COt =n∑
i=1
kupi oi,t +
n∑
i=1
kdowni vi,t
PRt = pec,t
m∑
j=1
PIOj,t
(zdRes
j,t Rdj,t + zcRes
j,t Rcj,t
)
PSt =m∑
j=1
PIOj,tLMPj,tPdisj,t
where PIOj,t = 1 if the PEV aggregator j participates in thegrid operation during t; PIOj,t = 0 if the PEV aggregator j isincluded in the base load instead of being dispatched during t.
The optimization is subject to the following constraints,where i = 1, . . . , n; j = 1, . . . , m; and t = 1, . . . , T .
1) Active power balance during tn∑
i=1
ui,tPi,t +m∑
j=1
(Pdis
j,t − Pchgj,t
)− Pload
t = 0. (2)
2) Operating and ramping limits of each generator
ui,tPi ≤ Pi,t ≤ ui,tPi (3)
Rdowni ≤ Pi,t − Pi,t−1 ≤ Rup
i . (4)
3) Minimum up and down time limits of generators [20]
− ui,t−1 + ui,t − ui,k ≤ 0, k = t, . . . , Hupi + t − 1 (5)
ui,t−1 − ui,t + ui,k ≤ 1, k = t, . . . , Hdowni + t − 1 (6)
−ui,t−1 + ui,t − oi,t ≤ 0 (7)
ui,t−1 − ui,t − vi,t ≤ 0. (8)
4) Reserve requirement
SR ≤n∑
i=1
PORi,t(min
(ui,t
(Pi − Pi,t
), ui,tτRup
i
))
+m∑
i=1
PIOj,t
(zdRes
j,t Rdj,t + zcRes
j,t Rcj,t
). (9)
5) Logical relations of the status of charging, discharging,and providing reserve of each PEV aggregator
zchgj,t + z
dis
j,t≤ 1 (10)
zchgj,t + z
dRes
j,t≤ 1 (11)
zdisj,t + z
cRes
j,t≤ 1 (12)
zdResj,t + z
cRes
j,t≤ 1. (13)
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
4 IEEE TRANSACTIONS ON SMART GRID
6) Charging and discharging power limits of eachaggregator
0 ≤ Pchgj,t ≤ PIOj,tz
chgj,t P
chgj,t (14)
0 ≤ Pdisj,t ≤ PIOj,tz
disj,t P
disj,t . (15)
7) Charging and discharging energy limits of eachaggregator
0 ≤ Pchgj,t σ t ≤ PIOj,tz
chgj,t
(wSoC
j,t − wSoCj,t
)(16)
0 ≤ Pdisj,t σ t ≤ PIOj,tz
disj,t wSoC
j,t (17)
where the term (wSoCj,t - wSoC
j,t ) represents the availablecharging capacity of the PEV aggregator j during t.
8) Reserve capacity limits of aggregator j in the dischargingmode during t
0 ≤ Rdj,t ≤ PIOj,t min
(P
disj,t − Pdis
j,t ,wSoC
j,t
σ t− Pdis
j,t
)(18)
where wSoCj,t /σ t represents the maximum discharging
power limited by the available energy capacity of thePEV aggregator j during t; min(P
disj,t − Pdis
j,t , (wSoCj,t /σ t)−
Pdisj,t ) represents the available reserve capacity provided
by the PEV aggregator j during t when it is in thedischarging status.
9) Reserve capacity limits of aggregator j in the chargingmode during t
0 ≤ Rcj,t ≤ PIOj,tP
chgj,t . (19)
10) Minimum requirement of regulation reserve of each PEVaggregator in the discharging status
zdResj,t = 0 if min
(P
disj,t − Pdis
j,t ,wSoC
j,t
σ t− Pdis
j,t
)< Rmin.
(20)
11) Minimum requirement of regulation reserve of each PEVaggregator in the charging status
zcResj,t = 0 if Pchg
j,t < Rmin (21)
zcResj,t = 0 if Pchg
j,t + wSoCj,t
σ t> wSoC
j,t . (22)
12) Energy balance of each PEV aggregator over T
T∑
t=1
ηchgj P
chg
j,tσ t = ECon
j +T∑
t=1
ηdisj P
dis
j,tσ t. (23)
The solution to the proposed model (1)–(23) provides thegrid operator with the optimal operation schedules for the ther-mal generators and PEV aggregators over the dispatch period.However, some parameters of this model are uncertain, suchas the available energy capacity of each PEV aggregator dueto the stochastic behaviors of PEVs. Therefore, the solution tothe DO model (1)–(23) will not be optimal. In this paper, anRO model is proposed to handle the parameter uncertainties.
C. Uncertainty Set
The uncertain parameters in the DO model (1)–(23) includethe available energy capacity of each PEV aggregator (wSoC
j,t ),reserve capability unit prices (pec,t), LMPs (LMPj,t), and sys-tem base load (Pload
t ) during each t. The values of pec,t, LMPj,t,and Pload
t can be forecasted with acceptable accuracy by usinga time series prediction method, such as the autoregressiveintegrated moving average (ARIMA) method [21], owing tothe availability of abundant historical data. However, it isunsuitable to use any time series prediction method to pre-dict wSoC
j,t and its level of uncertainty is much higher thanthose of pec,t, LMPj,t, and Pload
t in this paper because no his-
torical data of wSoCj,t are currently available. Therefore, for the
sake of simplicity, in this paper, only the available energycapacity of each PEV aggregator during t is considered anuncertain parameter, which is denoted as wSoC
j,t . However, theuncertainties of pec,t, LMPj,t, and Pload
t can be considered inthe same way.
The uncertainty set [12] RSoCj of wSoC
j,t for the aggregator jin a given dispatch period T is defined as follows:
RSoCj
(α, �j, wSoC
j,t
)
:={
wSoCj,t :∃γj,t ∈ R
T s.t. wSoCj,t ∈
[wSoC
j,t − αwSoCj,t
∣∣γj,t∣∣,
wSoCj,t + αwSoC
j,t
∣∣γj,t∣∣],
∣∣γj,t∣∣ ≤ 1,
1
T
T∑
t=1
∣∣γj,t∣∣ ≤ �j
}. (24)
The set RSoCj considers all possible available energy capac-
ity values of the PEV aggregator j in the range [wSoCj,t −
αwSoCj,t |γj,t|, wSoC
j,t + αwSoCj,t |γj,t|] with the constraint that the
average degree of uncertainties of wSoCj,t over the dispatch
period T is no more than �j. Thus, the value of α · �j repre-sents the uncertainty level of RSoC
j , where a larger value ofα · �j indicates a higher level of uncertainty.
Since |γj,t| ≤ 1 and 1/T‖γj,t‖1 ≤ �j, when �j = 0, theset (24) is a singleton {wSoC
j,t , j = 1, . . .m; t = 1, . . . , T},which corresponds to the nominal deterministic case. As �j
increases, the range of the uncertainty set RSoCj enlarges. It
means that a larger deviation of the total available energycapacity from the predicted value is considered for the PEVaggregator j. Then, the resulting RO model is more conser-vative, and the system is protected against a higher degree ofuncertainty. When �j = 1, RSoC
j only depends on wSoCj,t and α.
D. RO Model Formulation
Define x = [Pi,t, ui,t, oi,t, vi,t, Pchgj,t , zchg
j,t , Pdisj,t , zdis
j,t , zdResj,t ,
zcResj,t ](i = 1, . . . , n; j = 1, . . . , m; t = 1, . . . , T) the vector of
decision variables, where Pi,t, Pchgj,t , and Pdis
j,t are continuousvariables, and the others are binary (0/1) variables. The DOmodel (1)–(23) can be expressed in the following form:
minx
f (x)
s.t g(x) � 0 (25)
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
BAI AND QIAO: RO FOR BIDIRECTIONAL DISPATCH COORDINATION OF LARGE-SCALE V2G 5
where � means that there are both inequality and equalityconstraints.
Then, define w = [γj,t]( j = 1, . . . , m; t = 1, . . . , T), theset of uncertainty parameters and RSoC = ∏m
j=1 RSoCj , the
following RO model [12] can be formulated based on (25):
minx
maxw
f (x, w)
s.t. g (x, w) � 0 ∀w ∈ RSoC (26)
where wSoCj,t in (16)–(18), (20), and (22) is replaced with wSoC
j,t .The RO model (26) takes into account all possible scenarios
of wSoCj,t using the uncertainty set RSoC
j . The solution of (26) istherefore feasible and robust for any realization of the uncer-tain available energy capacities of the PEV aggregators. Incontrast, the solution of the DO model (25) only guarantees thefeasibility for a single nominal realization of wSoC
j,t ; while thestochastic optimization method only considers a finite numberof scenarios of wSoC
j,t .
III. SOLUTION METHOD TO THE RO MODEL
The robust counterpart approach [22] is applied to solvethe proposed RO model. The paradigm is to convert the ROmodel (26) to a computable model called the robust coun-terpart by removing the uncertainties using the method suchas explicit maximization, duality properties, or relaxation. Theexplicit maximization method is used to remove the uncertain-ties in this paper because (24) belongs to a typical norm-ballconstraint.
The constraints of the problem (26) involving uncertaintiesare the charging/discharging energy and regulation require-ment limits of each PEV aggregator below. They are notcomputable
0 ≤ Pchgj,t σ t ≤ PIOj,tz
chgj,t
(wSoC
j,t − wSoC
j,t
)(27)
0 ≤ Pdisj,t σ t ≤ PIOj,tz
disj,t wSoC
j,t (28)
0 ≤ Rdj,t ≤ PIOj,t min
(P
disj,t − Pdis
j,t ,wSoC
j,t
σ t− Pdis
j,t
)(29)
zdResj,t = 0 if min
(P
disj,t − Pdis
j,t , wSoCj,t /σ t − Pdis
j,t
)< Rmin (30)
zcResj,t = 0 if Pchg
j,t + wSoCj,t
σ t> wSoC
j,t . (31)
According to (24), the constraint (27) can be represented bya linear constraint (32) with a one-norm-bounded uncertainty
Pchgj,t σ t − PIOj,tz
chgj,t
(wSoC
j,t − wSoCj,t − αγj,tw
SoCj,t
)≤ 0
∀γj,t : |γj,t|1 ≤ 1 (32)
where Pchgj,t , zchg
j,t ∈ x, γj,t ∈ w, j = 1, . . . , m, and t =1, . . . , T .
The uncertainty parameters in the constraint (32) canbe removed by using the fact that max|γj,t|≤p wSoC
j,t γj,t is‖wSoC
j,t ‖p∗, where ‖·‖p∗ denotes the dual p-norm, and 1/p +
1/p∗ = 1 [23]. In this case, p is 1. Therefore, ‖·‖p∗ is the∞-norm. Thus, the constraint (27) is replaced with
Pchgj,t σ t − PIOj,tz
chgj,t
(wSoC
j,t − wSoCj,t − α
∥∥∥wSoCj,t
∥∥∥p∗
)≤ 0 (33)
where Pchgj,t , zchg
j,t ∈ x, j = 1, . . . , m, and t = 1, . . . , T .Constraint (33) is linear without uncertainty parameters.Hence, the constraint (27) with uncertainty parameters isreplaced with the computable constraints (33) without uncer-tainty parameters.
Similarly, the constraint (28)–(31) can be replaced withfour linear constraints (34)–(37), respectively, without anyuncertainty as well
Pdisj,t σ t − PIOj,tz
disj,t
(wSoC
j,t − α
∥∥∥wSoCj,t
∥∥∥p∗
)≤ 0 (34)
0 ≤ Rdj,t ≤ PIOj,t
min
(P
disj,t − Pdis
j,t
(wSoC
j,t − α
∥∥∥wSoCj,t
∥∥∥p∗
) /σ t − Pdis
j,t
)(35)
zdResj,t = 0 if
min
(P
disj,t − Pdis
j,t
(wSoC
j,t − α
∥∥∥wSoCj,t
∥∥∥p∗
) /σ t − Pdis
j,t
)< Rmin
(36)
zcResj,t = 0 if Pchg
j,t +wSoC
j,t − α
∥∥∥wSoCj,t
∥∥∥p∗
σ t> wSoC
j,t . (37)
This completes the process of removing the uncertainties ofthe RO model (26) and deriving its robust counterpart. TheRO problem (26) is therefore reformulated in the followingform:
minx
f (x)
s.t g(x) � 0. (38)
The objective f and constraints g of the new computablemodel (38) do not contain any uncertainty. Furthermore, thismodel is a mixed-integer quadratic programming problem,which can be solved by CPLEX or Gurobi efficiently.
IV. NUMERICAL SIMULATIONS
A. Simulation System Setup
A ten-generator power grid in [19] is used for simulationstudies. The fuel costs and parameters of the thermal gen-erators can be found in [19]. The total installed generationcapacity of the system is 1662 MW. The duration of eachdispatch interval t is 1 h and the value of T is 24. The 24-hbase load curve without PEVs is obtained by scaling the actualhourly loads of the Pennsylvania-New Jersey-Maryland (PJM)interconnection market on July 19th, 2013 [24] with the peakload of 1500 MW in the 12th hour and the valley load of1107 MW in the 24th hour. Seven generators (units 2–8)provide the reserve service. The reserve requirement is set
as = 3 × (maxt∈T(Ploadt + ∑m
j=1 (Pchgj,t − Pdis
j,t )) )1/2
[25]. Rmin
is set as 1 MW. Meanwhile, the power grid should provide30 763 MWh during the day without the PEV loads. The pro-posed RO model is validated and compared with the DO modelthrough a two-stage process as follows.
Stage 1: Obtain the DA generation and reserve schedulesfor the generators and PEV aggregators using the DO and ROmodels, respectively. In this stage, the ARIMA model is used
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
6 IEEE TRANSACTIONS ON SMART GRID
Fig. 1. Curves of base load as well as the capability unit price of reserveand LMP in the DA and RT markets.
to predict the DA hourly regulation market capability pricesand LMPs for all PEV aggregators based on the historical data.
Stage 2: The DA schedules obtained from the DO andRO models in stage 1 are dispatched on the next day ofoperation. The generators and PEV aggregators participate inthe real-time (RT) balancing market to sell/buy the deviatedenergy/reserve caused by the uncertain available energy capac-ities of the PEV aggregators in stage 1. The extra cost incurredin the RT market is calculated by using actual hourly regu-lation market capability prices and weighted-average LMPson the day of operation obtained from the PJM market. Theextra cost is then added to the total cost of the system describedby (1) for the validation and comparison of the DO and ROmodels.
Fig. 1 shows the curves of the base load, the predicted, andactual capability unit prices of the reserve and LMPs in theDA markets, and the actual capability unit price of the reserveand LMP in the RT market. It can be seen that the predictedDA prices match the actual DA prices with good precision.Furthermore, the RT prices are higher than the correspondingDA prices during most time of the day.
The PEVs in the system are grouped into three aggrega-tors according to their locations based on a survey on thestate of New York [26], where aggregators 1–3 cover the areaswith the populations less than 5 00 000, between 5 00 000 and3 million, and more than 3 million, respectively. Furthermore,it is assumed that within the three PEV aggregators, the charg-ing/discharging service can be provided to maximally 1400,52 000, and 85 000 PEVs, respectively, due to the capacitiesof the charging/discharging facilities. The parameters of thethree PEV aggregators are listed in Table I. The three PEVaggregators participate in the grid operation during 1:00–3:00,7:00–15:00, and 19:00–24:00, and their PIO values are set tobe one in these hours.
B. Construct Uncertainty Set
Since the historical data of wSoCj,t are not available, it is
unsuitable to use the ARIMA model or other time series pre-diction methods to predict it. In this paper, the values of wSoC
j,tare obtained by using a Monte Carlo simulation approach [27]based on the stochastic mobility behaviors of the PEVs inthe aggregator provided by the survey [26], which include the
TABLE IPARAMETERS OF THE THREE PEV AGGREGATORS
Fig. 2. Stochastic mobility behaviors of the three PEV aggregators: distri-butions of (a) departure time of daily commutation, (b) daily driving time,and (c) daily driving distance.
probabilities of departure time [Fig. 2(a)] for commutation,driving time [Fig. 2(b)], and driving distance [Fig. 2(c)]. Then,the numbers of PEVs parked in an aggregator during each tcan be determined using their departure and driving times.Specifically, in each t, the Monte Carlo method is used to sta-tistically generate a certain number (e.g., 500 in this paper) ofsamples. The value of wSoC
j,t is then calculated to be the aver-age value of the samples. Then, the uncertainty set RSoC
j isconstructed using (24). The upper and lower bounds of wSoC
j,tof each aggregator are set as 90% and 30% of its maximumenergy capacity during each t, respectively, by considering theallowable range of the state of charges of the PEVs’ batteriesduring normal operation.
As shown in Fig. 2, the mobility behaviors of the threePEV aggregators are similar. Their departure time distribu-tions all peak around 5:00 and 17:00. Most PEVs are parkedfrom 22:00 to 2:00 of the next day, and around 12:00. Thesedistribution curves coincide with the daily load curve, namely,most PEVs are parked during the peak-load and valley-loadperiods within a day. Therefore, it is preferable to chargePEVs during the valley-load period (from 23:00 to 2:00 of thenext day), and discharge PEVs during the peak-load period(around 12:00) if necessary. Furthermore, a PEV aggregatorcan participate in the grid operation to supply load when it isin the discharging mode, or provide regulation reserve whenit is in the charging or discharging mode.
According to (24) and (27)–(31) and the parameters ofeach PEV aggregator in Table I, the uncertainty set (the pre-dicted value and upper and lower bounds) of the available
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
BAI AND QIAO: RO FOR BIDIRECTIONAL DISPATCH COORDINATION OF LARGE-SCALE V2G 7
Fig. 3. Uncertainty set of the available reserve capacity of aggregator 1.
reserve capacity of each aggregator can be obtained fromRSoC
j . As an example, Fig. 3 shows the uncertainty set of theavailable reserve capacity of the PEV aggregator 1, where thestochastic mobility behavior of every PEV in this aggregatoris assumed to follow the standard uniform distribution, Rmin
is set to be 1 MW, and α = 0.1. The curve with stars corre-sponds to �j = 0; and the area between the two bounds coversall possible values of the available reserve capacity when�j = 1. In general, a larger α leads to a wider area betweenthe two bounds. However, when the low bound of the area islower than Rmin, the available reserve capacity will be empty,such as during 6:00–8:00 in Fig. 3, which means aggregator1 will not participate in the regulation reserve service duringthese hours.
C. Case Study
The proposed RO model is programmed in MATLAB 2012aand solved using Gurobi 5.1 on a 3.4 GHz desktop com-puter with a 16-G RAM. The relative gap between the lowerand upper objective bounds of Gurobi is set to be 1%. Inthis paper, the average combined fuel economy of PEVsis 30 kWh/100 miles [2]. Therefore, the daily energy con-sumption ECon
j of each PEV aggregator j can be estimatedaccording to the driving time [Fig. 2(b)] and driving dis-tance [Fig. 2(c)], and the results are 8.4, 213, and 510 MWhfor the PEV aggregators 1–3, respectively. Each of them isroughly 20% of the maximum available energy capacity of thePEV aggregator.
1) Model Validation in Stage 1: Fig. 4 compares the loadprofiles for four cases in stage 1 of validation: the base loadcase (no PEVs), the case when the DO model is used, andthe cases when the RO model with α = 0.1 and �j = 0.5or �j = 1 are used, where j = 1, 2, 3. In the DO andRO cases, the discharging power from the PEV aggregatorsis considered as a negative load. The differences between thebase load curve and three other curves indicate that the PEVaggregators are charged during the valley hours, which are1:00–3:00 and 19:00–24:00, in the DO or RO cases. In somevalley hours, e.g., 5:00–7:00, the PIO statuses of the PEVaggregators are set to be zero because most PEVs depart dur-ing those hours according to Fig. 2(a) and, therefore, cannotbe charged or participate in the grid operation. Furthermore,since the PEV aggregators in the DO and RO cases providedischarging power to the grid during 11:00–12:00, the peakload is reduced. Compared to the RO cases, the PEV aggrega-tors provide more discharging power in the DO case, which isanticipated in stage 1 DA scheduling because the predicted
Fig. 4. Comparison of load profiles for four cases during 24 h.
Fig. 5. Comparison of the itemized costs and the objective function valuesof the DO and RO cases.
available energy capacity of V2G is assumed to be accu-rate in the DO model but has uncertainty in the RO model.Moreover, in the RO cases, the discharging power reduces withthe increase of the uncertainty level. These results clearly showthe benefits of the V2G as a consequence of using the bidirec-tional dispatch coordination: the extra cost of using expensivegeneration to meet the peak load demand is reduced; and theramp-up/down costs of thermal generators are also reducedbecause the load curves are flattened by V2G.
2) Comparison of the DO and RO Models: Fig. 5 comparesthe itemized and total costs of the DO case and the two ROcases in stage 1, where the predicted available energy capaci-ties of the PEV aggregators used in the DO case are assumedaccurate. Owing to the fast response and no extra costs forramp-up/down, the PEV aggregators (other than extra high-cost generators) are selected to fulfill the spinning reserverequirement. As Fig. 5 shows, the objective function val-ues of the RO cases are slightly higher than that of theDO case.
However, the operation schedules obtained from the DOmodel are risky when the available reserve capacities of somePEV aggregators suffer negative forecast errors such that theycannot provide the scheduled spinning reserve capacities ordischarging power during some hours in the next day of actualdispatch. In this case, the PEV aggregators will participate inthe RT balancing market to compensate their energy/reservedeviations, as described in stage 2. This may lead to anincrease of the actual total system cost (i.e., the objectivefunction value) in the next day of operation.
Compared to the DO model, since the uncertainties havebeen considered in the RO model, less adjustment to theDA schedules is needed in the RT market. Therefore, theobjective function values of the RO cases in the next day ofoperation are less than that of the DO case. Fig. 6 showsthe percentage increase of the objective function value of theDO model in the next day of operation (stage 2) against that
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
8 IEEE TRANSACTIONS ON SMART GRID
Fig. 6. Percentage increase of the objective function value of the DO modelagainst that of the RO model versus the forecast error of the total availablereserve capacity of the PEV aggregators in the day of operation.
(a) (b)
Fig. 7. Adjusted reserve schedules in stage 2 as well as the costs of reserve indifferent stages obtained from (a) DO model and (b) RO model with α = 0.1and �j = 0.5 during 8:00–12:00 when there is −5% forecast error in the totalavailable reserve capacity of the PEV aggregators.
of the RO model versus the negative forecast error of thetotal available reserve capacity of the PEV aggregators from0 to 10% (with an increment of 1%), where the RO model(α = 0.1) with �j = 0.5 and �j = 1 (j = 1, 2, 3) has con-sidered the forecast error in the range of ±5% and ±10%,respectively. The objective function value of the DO model isslightly lower than that of the RO model if there is no fore-cast error, but is 4.5% more than that of the RO model evenin the case of −1% forecast error. Since a forecast error isinevitable, the RO model is robust and more economic thanthe DO model for scheduling the operations of the generatorsand PEV aggregators.
As an example, Fig. 7 compares the adjusted reserve sched-ules in stage 2 as well as the costs of reserve provided by thePEV aggregators in stages 1 and 2 obtained from the DOmodel and the RO model with α = 0.1 and �j = 0.5 dur-ing 8:00–12:00 when there is −5% forecast error in the totalavailable reserve capacity of the PEV aggregators. When theDO model is used, since the uncommitted generators havehigh start-up cost and/or slow response, the PEV aggrega-tors commit to provide the reserve capacity. As a result, thetotal reserve capacity provided by both generators and PEVaggregators in the next day of operation is higher than theactual reserve requirement due to the negative forecast errorof the total available reserve capacity of the PEV aggre-gators, leading to an increase of the spinning reserve cost,
Fig. 8. Comparison of the objective function values for different budgets ofuncertainty when the PEV mobility behaviors are modeled by uniform andnormal distributions.
as shown in Fig. 7(a). In contrast, since the RO model has cov-ered the −5% forecast error, no reserve adjustment is neededin stage 2, and the resulting scheduled reserve capacity inthe next day of operation is close to the requirement, whichavoids the increase of the spinning reserve cost, as shownin Fig. 7(b).
3) Robustness to the Probability Model of the PEVMobility: The probability distribution model of the stochasticmobility behaviors of the PEVs in a PEV aggregator can affectthe prediction accuracy of the PEV aggregator’s availableenergy capacity. It is desired that the RO model is robust to theprobability distribution model of the PEV mobility. To evaluatesuch robustness of the RO model, the Monte Carlo simulationapproach described in Section IV-B is used to obtain two dif-ferent uncertainty sets of the hourly available energy capacityfor each PEV aggregator on July 19, 2013 by assuming thatthe mobility of the PEVs follows the commonly used stan-dard uniform and normal distributions, respectively. For thenormal distribution, the mean of the hourly available energycapacity of each aggregator is set to be 70% of the maximumenergy capacity of the PEV aggregator during the hour, andthe standard deviation is set as 4% of the mean. The processof generating the uncertain sets of wSoC
j,t has been discussed inSection IV-B, where the value of α is chosen to be 0.1.
Fig. 8 compares the objective function values of using twodifferent distributions to model the stochastic PEV mobilitywhen α = 0.1 and �j increases from 0 to 1 with an increment of0.1. The maximum difference of the objective function valuesof the two cases is only 0.36%.
When using a certain probability distribution to model thePEV mobility, the parameters of the distribution may alsoaffect the prediction accuracy of the PEV aggregator’s avail-able energy capacity. In this paper, the mean of the normaldistribution is set to be 70% of the maximum energy capacityof each PEV aggregator in each hour, but 11 different standarddeviations from 0% to 40% of the mean value with an incre-ment of 4% are chosen for the Monte Carlo simulation. Usingthe aforementioned process, 11 different uncertainty sets ofthe hourly available energy capacity are obtained for each PEVaggregator. Then, the RO model with α = 0.1 and �j changingfrom 0 to 1 with an increment of 0.1 is simulated to obtainthe operation schedules of the generators and PEV aggregatorsusing each uncertainty set. The objective function value of thenext day of operation obtained in stage 2 versus the standarddeviation of the PEV mobility distribution and the budget ofuncertainty is shown in Fig. 9. The maximum difference of the
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
BAI AND QIAO: RO FOR BIDIRECTIONAL DISPATCH COORDINATION OF LARGE-SCALE V2G 9
Fig. 9. Objective function value of the RO model versus standard deviationof the normal PEV mobility distribution and budget of uncertainty.
Fig. 10. Comparison of objective function values of the RO model versusbudget of uncertainty at different levels of uncertainty.
objective function values for different uncertainty sets underthe same budge of uncertainty is only 0.01%.
The results in Figs. 8 and 9 indicate that the uncertaintiesof the PEV aggregators’ available energy capacities caused bythe uncertain PEV mobility have been well covered by theproposed RO model, regardless what probability distributionmodels and parameters are used to model these uncertainties.Thus, the proposed RO model (20) is robust to the probabilitydistribution modeling of the PEV mobility.
4) Effects of the Parameters of Uncertainty: Fig. 10 illus-trates the curves of the objective function value obtained fromthe RO model versus �j changing from 0 to 1 with an incre-ment of 0.1 when the value of α is 0.1, 0.3, and 0.5. Whenα = 0.1 or 0.3, the objective function value increases slowlyas �j increases. Even when �j = 1, which indicates a largeprediction error, the objective function value increases no morethan 0.19% and 1.22%, respectively, with respect to that when�j = 0. However, this increase reaches 3.1% when α = 0.5.The results show that by coordinating the charging and dis-charging operations of the PEV aggregators, the uncertainbehaviors of the PEVs have a low impact on the operationcost of the power grid when the uncertainty level of the PEVmobility is moderate, e.g., less than 0.3. Such an impact canbe further mitigated by improving the prediction accuracy inpractice. The results also reveal that for the same uncertaintylevel (i.e., α ×�j is constant), it is better to use a lower α andhigher �j than using a higher α and lower �j.
5) Impact of the Uncertain Price Volatility: Since the pro-posed RO model only considers the uncertainty of wSoC
j,t ,it is worth to investigate the impact of other uncertainties,such as the uncertain volatilities of the reserve capability unitprices (pec,t) and LMPs (LMPj,t), on the dispatch result. Twotests are performed to investigate the impact. In the first test,
Fig. 11. Relative differences of the objective function values of the casesin the second test with respect to that of the base case versus budget ofuncertainty �j for three different values of α.
the proposed RO model with the actual pec,t and LMPj,t issolved and the resulting optimal solution and objective func-tion value reflect the case when there is no uncertain volatilityin the prices, which is called the base case. In the secondtest, the proposed RO model with the predicted pec,t andLMPj,t is solved for different uncertain levels (i.e., differentcombinations of α and �j values) of wSoC
j,t . Then, the objec-tive function values of the RO model are recalculated usingthe actual pec,t and LMPj,t for different cases to represent thedispatch results when the uncertain volatilities of the pricesare considered. The relative differences in percentage of theobjective function values of the cases in the second test withrespect to that of the base case are plotted as functions of�j for three different α values in Fig. 11, where �j changesfrom 0 to 1 with an increment of 0.1. The results indicatethat the relative differences of the objective function valuesare less than 0.5% in most cases with price uncertainties. Themaximum relative difference between the two tests is only0.82%. Thus, the impact of the uncertain price volatilities ismuch less than that of the uncertain PEV mobility as shownin Fig. 6 when the DO model is used. Nevertheless, as dis-cussed in Section II-C, the uncertainties of the prices can beconsidered in the same way as the uncertainty of wSoC
j,t in theproposed RO model if necessary.
6) Benefit of the RO Model to the PEV Owners: The resultsin Sections IV-C2-Section IV-C5 have demonstrated the bene-fit of the RO model in minimizing the total cost of the systemfrom the perspective of the grid operator. However, the satis-faction of the PEV owners in an aggregator is also important toevaluate the benefit of the RO model. The PEV owners will besatisfied if they meet their charging requirement while gainingas much revenues as possible from participating in the powergrid operation. The revenues of the PEVs are considered asa kind of operation cost in the proposed DO and RO models,which are PRt and PSt in the objective function (1). However,if a PEV aggregator cannot provide all of the reserve capac-ity scheduled in the DA market (stage 1), it will have to buythe deviated reserve capacity in the RT market (stage 2) fromother providers, such as expensive generators. This will causean extra cost or a loss of the revenue for the PEV aggregator.The extra cost can be reflected by the variation of CRt + COt
between stages 1 and 2.Fig. 12 compares PRt + PSt and CRt + COt separately in
different stages obtained from the DO and RO models, whereα = 0.1 and �j = 0.5 are used in the RO model and there
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
10 IEEE TRANSACTIONS ON SMART GRID
Fig. 12. Comparison of the revenue of the PEV aggregators and the cost ofthe generators in the two stages using the DO and RO models.
is a −5% forecast error in the total available reserve capacityof the PEV aggregators. The results show that the value ofPRt + PSt in stage 2 decreases 5.3% compared with that instage 1 when using the DO model, but remains the same whenusing the RO model. Meanwhile, the total cost of the thermalgenerators to provide the reserve capacity (CRt) and the startupand shutdown cost (COt) using the DO model increases 12.6%in stage 2, leading to revenue losses of the PEV aggregators.On the contrary, the total cost of CRt +COt remains the samein stages 1 and 2 when using the RO model. As a result, thetotal objective function value in stage 2 increased 4.6% com-pared with that in stage 1 when using the DO model, whichcoincides with the results in Fig. 6. These results demon-strate that the PEV aggregators are more satisfied by usingthe RO model than using the DO model because the ROmodel enables the PEV owners to gain more revenues in theRT market.
V. CONCLUSION
This paper has proposed an RO model for bidirectional dis-patch coordination of PEVs in a power grid. The RO modelhas taken into account the uncertainty of the stochastic mobil-ity of the PEVs and the logical relations among charging,discharging, and providing regulation reserve of each PEVaggregator. The proposed RO model has been reformulatedas a deterministic mixed-integer quadratic programming prob-lem, which can be solved efficiently by using a state-of-the-artsolver, such as CPLEX or Gurobi. Numerical experiments havebeen conducted for a power grid with ten thermal genera-tors and three PEV aggregators to evaluate the proposed ROmodel. Results have shown that optimally coordinating PEVcharging/discharging with thermal generators and the abilityof PEV aggregators to provide regulation reserve service arebeneficial to power grid operation in terms of eliminating theinfluence of uncertainty on the overall cost while satisfyingthe transportation requirement of the PEVs. Moreover, the pro-posed RO model is robust to the probability modeling of theuncertain PEV mobility.
REFERENCES
[1] W. Kempton and S. E. Letendre, “Electric vehicles as a new powersource for electric utilities,” Transp. Res. D Transp. Environ., vol. 2,no. 3, pp. 157–175, Sep. 1997.
[2] W. Kempton and J. Tomic, “Vehicle-to-grid power fundamentals:Calculating capacity and net revenue,” J. Power Sources, vol. 144, no. 1,pp. 268–279, Jun. 2005.
[3] W. Kempton and J. Tomic, “Vehicle-to-grid power implementation: Fromstabilizing the grid to supporting large-scale renewable energy,” J. PowerSources, vol. 144, no. 1, pp. 280–294, Jun. 2005.
[4] C. D. White and K. M. Zhang, “Using vehicle-to-grid technology for fre-quency regulation and peak-load reduction,” J. Power Sources, vol. 196,no. 8, pp. 3972–3980, Apr. 2011.
[5] Y. Ma, T. Houghton, and A. Cruden, “Modeling the benefits of vehicle-to-grid technology to a power system,” IEEE Trans. Power Syst., vol. 27,no. 2, pp. 1012–1020, May 2012.
[6] E. Sortomme and M. A. El-Sharkawi, “Optimal combined bidding ofvehicle-to-grid ancillary services,” IEEE Trans. Smart Grid, vol. 3, no. 1,pp. 70–79, Mar. 2012.
[7] S. Han, “Development of an optimal vehicle-to-grid aggregator for fre-quency regulation,” IEEE Trans. Smart Grid, vol. 1, no. 1, pp. 65–72,Jun. 2010.
[8] T. K. Kristoffersen, K. Capion, and P. Meibom, “Optimal charging ofelectric drive vehicles in a market environment,” Appl. Energy, vol. 88,no. 5, pp. 1940–1948, May 2011.
[9] A. Y. Saber, “Plug-in vehicles and renewable energy sources for costand emission reductions,” IEEE Trans. Ind. Electron., vol. 58, no. 4,pp. 1229–1238, Apr. 2011.
[10] M. W. Khalid, A. T. Al-Awami, and E. Sortomme, “Stochastic-programming-based bidding strategy for V2G services,” in Proc.4th IEEE/PES Innov. Smart Grid Technol. Eur., Lyngby, Denmark,Oct. 2013, pp. 1–5.
[11] M. E. Khodayar, L. Wu, and M. Shahidehpour, “Hourly coordination ofelectric vehicle operation and volatile wind power generation in SCUC,”IEEE Trans. Smart Grid, vol. 3, no. 3, pp. 1271–1279, Sep. 2012.
[12] A. Ben-Tal, L. E. Ghaoui, and A. Nemirovski, Robust Optimization.Princeton, NJ, USA: Princeton Univ. Press, 2009.
[13] S. Dehghan, N. Amjady, and A. Kazemi, “Two-stage robust genera-tion expansion planning: A mixed integer linear programming model,”IEEE Trans. Power Syst., vol. 29, no. 2, pp. 584–597, Mar. 2014.
[14] D. Bertsimas, E. Litvinov, X. A. Sun, J. Zhao, and T Zheng, “Adaptiverobust optimization for the security constrained unit commitment prob-lem,” IEEE Trans. Power Syst., vol. 28, no. 1, pp. 52–63, Feb. 2013.
[15] C. Lee, C. Liu, S. Mehrotra, and M. Shahidehpour, “Modeling trans-mission line constraints in two-stage robust unit commitment problem,”IEEE Trans. Power Syst., vol. 29, no. 3, pp. 1221–1231, May 2014.
[16] Y. Guan and J. Wang, “Uncertainty sets for robust unit commitment,”IEEE Trans. Power Syst., vol. 29, no. 3, pp. 1439–1440, May 2014.
[17] (Mar. 2014). IBM ILOG CPLEX 12.4. [Online]. Available:http://www-01.ibm.com/
[18] (Mar. 2014). Gurobi Optimizer 5.1. [Online]. Available:http://www.gurobi.com
[19] C.-P. Cheng, C.-W. Liu, and C.-C. Liu, “Unit commitment by Lagrangianrelaxation and genetic algorithms,” IEEE Trans. Power Syst., vol. 15,no. 2, pp. 707–714, May 2000.
[20] (Jun. 2005). Minimum Up/Down Polytopes of the Unit CommitmentProblem With Start-Up Costs, IBM Research Report. [Online]. Available:http://domino.research.ibm.com/library/cyberdig.nsf/papers/CDCB02A7C809D89E8525702300502AC0/$File/rc23628.pdf
[21] A. J. Conejo, M. Carrión, and J. M. Morales, Decision Making UnderUncertainty in Electricity Markets. New York, NY, USA: Springer, 2010.
[22] A. Ben-Tal and A. Nemirovski, “Robust optimization-methodology andapplications,” Math. Prog. vol. 92, no. 3, pp. 453–480, May 2002.
[23] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.:Cambridge Univ. Press, 2004.
[24] (Dec. 2013). PJM Energy Market Data. [Online]. Available:http://www.pjm.com/ markets-and-operations/energy.aspx
[25] Y. Rebours and D. Kirschen, “A survey of definitions and specificationsof reserve services,” Report, Univ. of Manchester, Oct. 2005.
[26] (Jun. 2012). New York State 2009 NHTS Comparison Report. [Online].Available: https://www.dot.ny.gov/divisions/policy-and-strategy/darb/dai-unit/ttss/repository/ComparisonRpt09-95-011107.pdf
[27] D. Dallinger, D. Krampe, and M. Wietschel, “Vehicle-to-grid regulationreserves based on a dynamic simulation of mobility behavior,” IEEETrans. Smart Grid, vol. 2, no. 2, pp. 302–313, Jun. 2011.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
BAI AND QIAO: RO FOR BIDIRECTIONAL DISPATCH COORDINATION OF LARGE-SCALE V2G 11
Xiaoqing Bai received the B.S. degree in softwareengineering and the Ph.D. degree in electrical engi-neering, in 1991 and 2010, respectively, both fromGuangxi University, Nanning, China.
She is currently a Post-Doctoral ResearchAssociate with the Department of Electrical andComputer Engineering, University of Nebraska–Lincoln, Lincoln, NE, USA. She was with GuangxiKey Laboratory of Power System Optimization andEnergy Technology, Guangxi University. Her currentresearch interests include optimization theories and
their applications in power systems.
Wei Qiao (S’05–M’08–SM’12) received the B.E.and M.E. degrees in electrical engineering fromZhejiang University, Hangzhou, China, in 1997and 2002, respectively; the M.S. degree in highperformance computation for engineered systemsfrom Singapore-MIT Alliance, Singapore; and thePh.D. degree in electrical engineering from theGeorgia Institute of Technology, Atlanta, GA, USA,in 2003 and 2008, respectively.
Since 2008, he has been with the University ofNebraska–Lincoln (UNL), Lincoln, NE, USA, where
he is currently an Associate Professor in the Department of Electricaland Computer Engineering. His current research interests include renew-able energy systems, smart grids, microgrids, condition monitoring, energystorage systems, power electronics, electric machines and drives, and com-putational intelligence. He has authored/co-authored three book chapters andover 140 papers in refereed journals and international conference proceedings,and has five international/U.S. patents pending.
Dr. Qiao was the recipient of the 2010 U.S. National Science FoundationCAREER Award, the 2010 IEEE Industry Applications Society (IAS) AndrewW. Smith Outstanding Young Member Award, the 2012 UNL College ofEngineering Faculty Research and Creative Activity Award, the 2011 UNLHarold and Esther Edgerton Junior Faculty Award, and the 2011 UNL Collegeof Engineering Edgerton Innovation Award. He was also the recipient of fourbest paper awards from IEEE IAS, Power and Energy Society, and PowerElectronics Society. He is an Associate Editor of the IEEE TRANSACTIONS
ON ENERGY CONVERSION, IET Power Electronics, and the IEEE JOURNAL
OF EMERGING AND SELECTED TOPICS IN POWER ELECTRONICS. He is theCorresponding Guest Editor of a special section on Condition Monitoring,Diagnosis, Prognosis, and Health Monitoring for Wind Energy ConversionSystems of the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS. Hewas also an Associate Editor of the IEEE TRANSACTIONS ON INDUSTRY
APPLICATIONS from 2010 to 2013.
